Rate-compatible protograph ldpc code families with linear minimum distance

ABSTRACT

Digital communication coding methods are shown, which generate certain types of low-density parity-check (LDPC) codes built from protographs. A first method creates protographs having the linear minimum distance property and comprising at least one variable node with degree less than 3. A second method creates families of protographs of different rates, all having the linear minimum distance property, and structurally identical for all rates except for a rate-dependent designation of certain variable nodes as transmitted or non-transmitted. A third method creates families of protographs of different rates, all having the linear minimum distance property, and structurally identical for all rates except for a rate-dependent designation of the status of certain variable nodes as non-transmitted or set to zero. LDPC codes built from the protographs created by these methods can simultaneously have low error floors and low iterative decoding thresholds, and families of such codes of different rates can be decoded efficiently using a common decoding architecture.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims priority to U.S. Provisional 60/931,442filed on May 23, 2007, which is incorporated herein by reference in itsentirety. The present application is also related to U.S. Pat. No.7,343,539 for “ARA Type Protograph Codes”, issued on Mar. 11, 2008, alsoincorporated herein by reference in its entirety.

STATEMENT OF GOVERNMENT GRANT

The invention described herein was made in the performance of work undera NASA contract, and is subject to the provisions of Public Law 96-517(35 USC 202) in which the Contractor has elected to retain title.

FIELD

The present disclosure relates to constructing low-density parity-check(LDPC) codes from small template graphs called protographs. More inparticular, it relates to new methods for designing rate-compatiblefamilies of protographs of different rates, all having the linearminimum distance property, as well as a new method for constructingindividual protographs having one or more degree-2 variable node yetguaranteed to possess the linear minimum distance property. LDPC codesof arbitrarily large size can be built by known methods expanding fromthese individual protographs or rate-compatible protograph families.

SUMMARY

According to a first aspect, a digital communication coding method isprovided, comprising: providing a low-density parity-check (LDPC) coderepresented by a protograph, the protograph being provided with a linearminimum distance property, the protograph comprising at least onevariable node with degree less than 3, and the protograph obtained byapplying one or more operations in succession starting with a baseprotograph having all variable node degrees at least 3, each said one ormore operations comprising a splitting of one check node of the baseprotograph or subsequent protographs into two check nodes and connectingsaid two check nodes with a transmitted degree-2 variable node.

According to a second aspect, a digital communication coding method isprovided, comprising: providing a family of low-density parity-check(LDPC) codes of different rates but constant input block size, saidfamily represented by a set of protographs of different rates, saidprotographs being provided with a linear minimum distance property, andsaid protographs obtained by applying one or more operations insuccession starting with a base protograph having all variable nodedegrees at least 3, each said one or more operations comprising asplitting of one check node of the base protograph or subsequentprotographs into two check nodes and connecting said two check nodeswith either a transmitted degree-2 variable node or a non-transmitteddegree-2 variable node.

According to a third aspect, a digital communication coding method isprovided, comprising: providing a family of low-density parity-check(LDPC) codes of different rates but constant output block size, saidfamily represented by a set of protographs of different rates, saidprotographs being provided with a linear minimum distance property, andsaid protographs obtained by applying one or more operations insuccession starting with a base protograph having all variable nodedegrees at least 3, each said one or more operations comprising aconnecting of two check nodes of the base protograph or subsequentprotographs with either a non-transmitted degree-2 variable node or anon-transmitted degree-2 variable node set to zero.

Further aspects of the present disclosure are described in the writtenspecification, drawings and claims of the present application.

DEFINITIONS

As known to the person skilled in the art and as also mentioned in U.S.Pat. No. 7,343,539 incorporated herein by reference in its entirety, alow-density parity-check (LDPC) code is a linear code determined by asparse parity-check matrix H having a small number of 1s per column. Thecode's parity-check matrix H can be represented by a bipartite Tannergraph wherein each column of H is represented by a transmitted variablenode, each row by a check node, and each “1” in H by a graph edgeconnecting the variable node and check node that correspond to thecolumn-row location of the “1”. The code's Tanner graph may additionallyhave non-transmitted variable nodes or state variables, which do notcorrespond to columns of the parity-check matrix but which can simplifythe graphical representations of some codes. Each check or constraintnode defines a parity check operation. Moreover, the fraction of atransmission that bears information is called the rate of the code. AnLDPC code can be encoded by deriving an appropriate generator matrix Gfrom its parity-check matrix H. An LDPC code can be decoded efficientlyusing a well-known iterative algorithm that passes messages along edgesof the code's Tanner graph from variable nodes to check nodes andvice-versa until convergence is obtained.

A protograph is a small bipartite template graph containing variablenodes joined by edges to check nodes. Arbitrarily large LDPC codes canbe constructed from any given protograph by first making an arbitrarynumber of copies of the protograph and then permuting the endpoints ofthe set of copies of a given edge among the set of copies of thevariable or check nodes at the endpoints of that edge in the protograph.This procedure of expanding a protograph to create a large LDPC code canbe performed in various ways. One method assigns a permutation,circulant, or circulant permutation of size N to each edge of theprotograph. If the protograph has n transmitted variable nodes, then thederived graph for the expanded LDPC code has nN transmitted variablenodes. Another method is useful when the protograph has parallel edges,which are two or more edges connecting the same pair of variable andcheck nodes. One can first use the copy and permute operation to expandthe protograph by a factor of N₁, to obtain a larger protograph withoutparallel edges. Then this method assigns a permutation, circulant, orcirculant permutation of size N₂ to each edge of the larger protographas expanded from the original protograph. The derived graph now hasnN₁N₂ transmitted variable nodes if the original protograph had ntransmitted variable nodes.

The rate of a protograph is defined to be the lowest (and typical) rateof any LDPC code constructed from that protograph. All LDPC codesconstructed from a given protograph have the same rate except forpossible check constraint degeneracies, which can increase (but neverdecrease) this rate and typically occur only for very small codes. Sincethe protograph serves as a blueprint for the Tanner graph representingany LDPC code expandable from that protograph, it also serves as ablueprint for the routing of messages in the iterative algorithm used todecode such expanded codes.

Excluding check nodes connected to degree-1 variable nodes, applicantshave proved that the number of degree-2 nodes should be at most one lessthan the number of check nodes for a protograph to have the linearminimum distance property. A given protograph is said to have the linearminimum distance property if the typical minimum distance of a randomensemble of arbitrarily large LDPC codes built from that protographgrows linearly with the size of the code, with linearity coefficientδ_(min)>0.

The iterative decoding threshold of a given protograph is similarlydefined with respect to this random ensemble of LDPC codes as the lowestvalue of signal-to-noise ratio for which an LDPC decoder's iterativedecoding algorithm will find the correct codeword with probabilityapproaching one as the size of an LDPC code built from that protographis made arbitrarily large. Iterative decoding thresholds can becalculated by using the reciprocal channel approximation. Thresholds canbe lowered either by using preceding or through the use of one veryhigh-degree node in the base protograph. A protograph is said to have alow iterative decoding threshold if its threshold is close to thecapacity limit for its rate.

A family of protographs of different rates is said to be rate-compatibleif the protographs for different rates are sufficiently similar to eachother in structure that a decoder for an entire family of LDPC codes ofdifferent rates built from these protographs can be implemented muchmore simply than separate decoders for each rate. The rate-compatiblefamilies described in the present disclosure are rate-compatible to themaximum extent, in the sense that the protographs for all rates have theidentical number of variable and check nodes and identical connectionsbetween variable and check nodes, with the only differences betweenprotographs of different rates being a trivial designation of the status(transmitted, non-transmitted, or set to zero) of some of the variablenodes.

The protograph-based LDPC codes according to the present disclosuresimultaneously achieve low iterative decoding thresholds, linear minimumdistance, and can provide various code rates. In addition the proposedcodes may have either fixed input block or fixed output block sizing andin both cases provide rate compatibility. In fact, one encoder and onedecoder can support different code rates. Several families are describedwhere each includes high- to low-rate codes and each has minimumdistance linearly increasing in block size with capacity-approachingperformance thresholds.

As will be described, the linear minimum distance property was verifiedby considering ensemble average weight enumerators wherein the minimumdistance of the proposed codes was shown to grow linearly with blocksize.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a degree-m check and its equivalent representation forpartial weight enumeration.

FIG. 2 shows an example of how to apply the method according to thepresent disclosure to construct rate-⅔ and rate-½ protographs startingfrom a rate-⅘ base protograph, for which a high-degree variable node wasused in the base protograph.

FIG. 3 shows a rate-compatible family of identically structuredprotographs generated from the same base protograph and for the samerates shown in FIG. 2, by applying the method according to the presentdisclosure.

FIG. 4 shows simulation results for LDPC codes with input block sizek=3680 expanded from the rate-½ through rate-⅘ protographs in FIG. 3.

FIG. 5 shows a rate-½ protograph with 7 variable nodes of degree 3 andone variable node of high degree 16, used to lower the iterativedecoding threshold.

FIG. 6 shows construction of a rate-⅞ protograph starting from therate-½ base protograph shown in FIG. 5, by applying the method accordingto the present disclosure.

FIG. 7 shows a table that specifies different combinations of status(set to zero or not) of the three non-transmitted variable nodes in FIG.6, that allow the single protograph structure in FIG. 6 to representprotographs of rates ½, ⅝, ¾, and ⅞.

FIG. 8 shows a rate-compatible family of identically structuredprotographs of rates ½, ⅝, ¾, and ⅞, generated from the singleprotograph structure in FIG. 6 according to the prescriptions in thetable in FIG. 7.

FIG. 9 shows the performance of LDPC codes with output block size n=5792expanded from the rate-½ through rate-⅞ protographs in FIG. 8.

FIG. 10 shows how to apply the method according to the presentdisclosure, starting from a base protograph with all variable nodedegrees 3 or higher, and including precoding to lower the iterativedecoding threshold, to obtain a rate-½ AR4JA protograph.

FIG. 11 shows protographs for a family of AR4JA protographs of rates ½through ⅞, together with the iterative decoding thresholds (in dB)achieved by this family compared to the corresponding capacity limits(in dB).

FIG. 12 shows bit and frame error rate simulation results for LDPC codeswith input block size k=4096 expanded from the rate-½ through rate-⅘AR4JA protographs in FIG. 11.

FIG. 13( a) shows an 8-node rate-½ protograph with all variable nodedegrees 3.

FIG. 13( b) shows the same protograph of FIG. 13( a), except with onevariable node degree increased to 16 to lower the iterative decodingthreshold.

FIG. 14 shows a protograph with 11 variable nodes of degree 3, and onevariable node of degree 18.

DETAILED DESCRIPTION

The codes constructed in accordance with the methods described in thepresent disclosure can fall into four categories. Each can findapplication in different scenarios. These four categories are: (1) LDPCcodes built from individual protographs having the linear minimumdistance property and having one or more variable nodes of degree-2; (2)Rate-compatible families of LDPC codes with fixed input block size builtfrom a family of protographs of different rates all having the linearminimum distance property, with protographs for all rates identical instructure except for the transmit status of degree-2 variable nodes; (3)Rate-compatible families of LDPC codes with fixed output block sizebuilt from a family of protographs of different rates all having thelinear minimum distance property, with protographs for all ratesidentical in structure except for the status of degree-2 variable nodes;(4) LDPC codes built from individual protographs having the linearminimum distance property and low iterative decoding threshold close tothe capacity limits for their respective rates.

1. Protographs Having the Linear Minimum Distance Property and HavingOne or More Variable Nodes of Degree-2.

In most wireless standards, either a convolutional or turbo code withpuncturing for various code rates is included. The proposed LDPC codeswith the linear minimum distance property have very low error floor, canproduce various code rates, and a single fast decoder using beliefpropagation can be implemented to handle the different code rates. Thusthese LDPC codes have all the required practical features, and theyachieve excellent performance, and very low error floor even for shortblocks. Low error floors are necessary in applications wherere-transmission is either impossible, for instance when broadcastingvideo from one point to many, or undesirable, for example very longdelay applications such as communicating with probes that are deep inspace. The use of degree-2 nodes to lower the iterative decodingthreshold is relevant to all communication scenarios in which eitherbandwidth or power efficiency are important. This is because lowerthresholds imply greater range for a given transmitted power or highersignaling rate for the same range and power.

Computation of ensemble weight enumerators for protograph LDPC codesrequires knowledge of the partial weight enumerator A_(w) ₁ _(, w) ₂_(. . . , w) _(m) for every check with degree m in the protograph. Anydegree-m check node can be split into an equivalent subgraph with twocheck nodes of degree m₁+1 and m₂+1 connected by a degree-2 variablenode, such that m₁+m₂=m.

FIG. 1 shows a degree-m check and its equivalent representation forpartial weight enumeration.

The partial weight enumerator for the check with degree m, expanded torepresent m binary sequences each of length N, is obtained from thepartial weight enumerators of the two checks with degrees m₁+1 and m₂+1as

$A_{w_{1},{w_{2}\mspace{11mu} \ldots}\mspace{14mu},w_{m}}^{c} = {\sum\limits_{l = 1}^{N}\frac{A_{w_{1},\mspace{11mu} \ldots \mspace{14mu},w_{m_{1}},l}A_{w_{{m_{1} + 1},\mspace{11mu} \ldots \mspace{14mu},w_{m},l}}}{\begin{pmatrix}N \\l\end{pmatrix}}}$

Applicants use this idea to construct protograph LDPC codes that includedegree-2 variable nodes to achieve good iterative decoding thresholds,yet also have minimum distance growing linearly with block size.

The starting point can be a high-rate protograph LDPC code where thedegrees of all variable nodes are at least 3. It is known that such acode ensemble has minimum distance that grows linearly with block size.An example of such a base protograph is the rate-⅘ protograph shown atthe top of FIG. 2.

Next, a check node in the protograph is split into two checks and thetotal number of edges into the original check is distributed between thetwo new checks. After that, these two checks are connected with anon-transmitted degree-2 variable node. The resulting protograph has oneadditional check node and one new non-transmitted (i.e. not output, orpunctured) degree-2 variable node. The corresponding protograph LDPCcode ensemble will have the same average weight enumerator, and so itsensemble minimum distance will grow linearly with block size with thesame linearity coefficient. The overall code rate remains the same. Thisoperation of splitting one check node into two and connecting theresulting two check nodes with a degree-2 variable node can be appliedagain to subsequent protographs that result from previous applicationsof this operation.

Finally, if the new degree-2 variable node is changed from anon-transmitted node to a transmitted node, a lower-rate protograph isobtained, but the property that the ensemble minimum distance growslinearly with block size will be preserved. FIG. 2 illustrates theapplication of the procedure described in [0023] to [0025] to obtainrate-⅔ and rate-½ protographs, starting from the rate-⅘ base protographshown at the top of FIG. 2.

2. Rate Compatible Families of Identically Structured Protographs withFixed Input Block Size all Having the Linear Minimum Distance Property.

Certain applications require rate compatible codes that have a fixedinput block size. This implies that only the number of parity bits isallowed to vary. Examples are the CCSDS standard for Deep Spacecommunications. Other examples include systems in which a higher layerproduces packets with a fixed size. In such instances it may bedesirable for the frame boundaries from the higher layer to coincidewith frame boundaries from the coding layer.

To design such codes for these applications, applicants start with ahigh-rate protograph LDPC code with variable node degrees of at least 3,as in [0023]. Lower rate codes are obtained by splitting check nodes andconnecting them with degree-2 nodes. Applicants have proven that thisguarantees that the linear minimum distance property is preserved forthe lower-rate codes.

Continuing with the method described in [0023] to [0025], at eachoperation comprising a splitting of one check node and connecting thetwo resulting nodes with degree-2 variable nodes, one has the option ateach step to designate the status of the connecting degree-2 node to beeither transmitting or non-transmitting. The rate of the protograph islowered each time the connecting node is designated as a transmittednode, while the rate is unchanged each time the connecting node isdesignated as a non-transmitted node. Starting with a base protograph ofhighest rate and continuing this process for S steps, each stepcomprising a splitting of one check node into two and connecting theresulting two check nodes with either a degree-2 transmitted variablenode or a degree-2 non-transmitted node, one obtains a rate-compatiblefamily of protographs, all of which have identical graph structureincluding S additional variable nodes and S additional check nodescompared to the numbers of these nodes in the base protograph, the onlydifference among rates being the transmit status (transmitted ornon-transmitted) of the S new degree-2 variable nodes.

By the process described in [0028], protographs for all rates areobtained by applying the same number of splitting and connectingoperations; the only rate-dependent difference is whether the connectingnode is transmitted or not. The lowest-rate protograph is obtained whenall of the degree-2 connecting nodes are transmitted, the highest-rateprotograph is obtained when all of the degree-2 connecting nodes arenon-transmitted, and intermediate-rate protographs are obtained whensome of the degree-2 connecting nodes are of each type. The resultingprotographs are identical in structure for all rates, except for thetransmit status (transmitted or non-transmitted) of the degree-2connecting nodes.

The three protographs shown in FIG. 3 are equivalent to those shown inFIG. 2 but are obtained from the rate-⅘ base protograph by the processdescribed in [0028]. The result is a set of protographs in FIG. 3 thatare identical in structure for all three rates, with the only differencebeing the transmit status of the three degree-2 connecting nodes. Therate-½ protograph is obtained when all three of these nodes aretransmitted, the rate-⅘ protograph results when none of these threenodes in transmitted, and the rate-⅔ code is obtained when one of thesethree nodes is transmitted.

Reversing the construction process, it can be noted that the higher-rateprotographs in FIG. 3 can be obtained by simply puncturing some of thedegree-2 nodes of the rate-½ protograph.

Note that the rate-½ protograph code in FIG. 2 or FIG. 3 has ensembleasymptotic minimum distance δmin=0.005. The resulting codes for rates ½,⅔, and ⅘ have a fixed input block size.

FIG. 4 shows simulation results for LDPC codes with input block sizek=3680 expanded from the rate-½ through rate-⅘ protographs in FIG. 3,for which a high-degree node was used in the base protograph.

3. Rate Compatible Families of Identically Structured Protographs withFixed Output Block Size all Having the Linear Minimum Distance Property.

Fixed output block length codes are desirable in scenarios where aframing constraint is imposed on the physical layer. Perhaps the mostcommon example is provides by links that employ orthogonal frequencydivision multiplexing (OFDM) as the modulation. Digital subscribed linemodems use such a modulation as does the WiMax standard.

In the present disclosure, applicants construct rate-compatibleprotograph-based LDPC codes for fixed block lengths that simultaneouslyachieve low iterative decoding thresholds and linear minimum distance(δ_(min)>0) such that error floors may be suppressed. These codes inaccordance with the present disclosure are constructed starting from alow-rate base protograph that has degree-3 nodes and one high-degreenode (which serves to lower the iterative decoding threshold). Toconstruct a rate-compatible protograph family with fixed output blocklength, one can combine check nodes of the low-rate protograph to formhigher-rate protographs. In the present disclosure, applicants usedegree-2 non-transmitted nodes to implement check node combining(thereby achieving rate compatibility), but use no transmitted degree-2nodes. The condition where all constraints are combined corresponds tothe highest-rate code.

By avoiding transmitted degree-2 nodes applicants obtain a family ofprotographs all of which are guaranteed to have the linear minimumdistance property. Limiting the protograph design to the use of degree-3and higher variable nodes is a sufficient but not necessary condition topreserve the linear minimum distance property. For example, a protographconstruction that uses transmitted degree-2 nodes and still achieveslinear minimum distance is described previously in [0022] to [0025].

The present disclosure describes small protograph-based codes thatachieve competitively low thresholds without the use of degree-2variable nodes and without degree-1 pre-coding in conjunction with thepuncturing of a high degree node. The iterative decoding threshold forproposed rate-½ codes are lower, by about 0.2 dB, than the best knownirregular LDPC codes with degree at least three. Iterative decodingthresholds as low as 0.544 dB can be achieved for small rate-½protographs with variable node degrees at least three.

Higher-rate codes are obtained by connecting check nodes with degree-2non-transmitted nodes. This is equivalent to constraint combining in theprotograph. The condition where all constraints are combined correspondsto the highest-rate code.

The 8-node rate-½ protograph in FIG. 5 can be used as a base protographto illustrate this construction. FIG. 6 shows the construction of arate-⅞ protograph starting with the rate-½ base protograph in FIG. 5.

The single protograph structure in FIG. 6 can be used for rates ½, ⅝, ¾,and ⅞ if the nodes 8, 9, and 10 are properly set to “0” bit (equivalentto not having degree-2 node connections), or “X” where “X” represents nobit assignment to the node (a non-transmitted node). The table in FIG. 7describes this operation. At the decoder the corresponding nodes to “0”bits are assigned highly reliable values, and to nodes “X” zeroreliability values. Iterative decoding thresholds computed for the rate½, ⅝, ¾, and ⅞ protographs are 0.618 (gap to capacity 0.43 db), 1.296(gap to capacity 0.48 dB), 1.928 (gap to capacity 0.30 dB), and 3.052 dB(gap to capacity 0.21 dB) respectively.

The rate-compatible family of protographs described by the masterprotograph in FIG. 6 together with the table in FIG. 7 is illustratedexplicitly in FIG. 8. This family of protographs is describedequivalently as follows. With each operation comprising a connecting oftwo check nodes with degree-2 non-transmitted variable nodes, one hasthe option at each step to designate the status of the connectingdegree-2 node to be either set to zero or not. The rate of theprotograph is unchanged each time the connecting node is set to zero,while the rate is increased each time the connecting node is not set tozero. Starting with a base protograph of lowest rate and continuing thisprocess for S steps, each step comprising a connecting of two checknodes with a non-transmitted degree-2 variable node either set to zeroor not, one obtains a rate-compatible family of protographs, all ofwhich have identical graph structure including S additional variablenodes and the same number of check nodes compared to the numbers ofthese nodes in the base protograph, the only difference among ratesbeing the status (set to zero or not) of the S new non-transmitteddegree-2 variable nodes.

By the process described in [0041], protographs for all rates areobtained by applying the same number of connecting operations; the onlyrate-dependent difference is whether the non-transmitted connecting nodeis set to zero or not. The highest-rate protograph is obtained when noneof the degree-2 connecting nodes are set to zero, the lowest-rateprotograph is obtained when all of the degree-2 connecting nodes are setto zero, and intermediate-rate protographs are obtained when some of thedegree-2 connecting nodes are of each type. The resulting protographsare identical in structure for all rates, except for the status (set tozero or not) of the non-transmitted degree-2 connecting nodes.

FIG. 9 shows bit (solid curves) and frame (dashed curves) error rateresults for LDPC codes with output block length n=5792 codes expandedfrom the rate-compatible protograph family of FIG. 8. The three lowestrates exhibit no error flooring at frame error rates of 3e-7 and higher.However the rate-⅞ code does display error events near the 1e-6 leveldue to certain graph anomalies. Note that the choices of circulantpermutations used to construct the codes of rates ½, ⅝, and ¾ were thesame as those used for the rate-⅞ code, with the exception of removal ofcirculants (and corresponding edges) associated with those nodes (8, 9,and/or 10) that are set to zero in FIG. 8. For an efficient commondecoder implementation, instead of being removed, these nodes could beassigned highly reliable values corresponding to “0” bits to cancel anycontributions to their respective check nodes.

4. Protographs Having the Linear Minimum Distance Property and LowIterative Decoding Threshold.

In applications where latency is not an issue long block length codescan be used to maximize power and or bandwidth efficiency. Protographcode thresholds can be minimized even further for these uses through theaddition of preceding structures. Direct Broadcast Satellite and DeepSpace applications would benefit from this technique.

FIG. 10 shows an example of the construction described in [0022] to[0025], as applied to obtain a rate-½ AR4JA protograph, starting with arate-⅔ base protograph having all variable node degrees 3 or higher. Inthe last step of this example, applicants also attached an accumulatoras a precoder to lower the iterative decoding threshold. The iterativedecoding threshold for this rate-½ code is 0.64 dB, and the asymptoticgrowth rate of the ensemble minimum distance is δ_(min)=0.015.

After using the check node splitting technique described in [0022] to[0025] to design a particular low-rate code such as the rate-½ AR4JAcode in FIG. 10 with minimum distance guaranteed to grow linearly withblock size, this property will be preserved if additional variable nodesof degree-3 and higher are attached to this low-rate protograph.

Thus, it can be concluded that the entire AR4JA family described in U.S.Pat. No. 7,343,539, incorporated herein by reference in its entirety,for rates r=(n+1)/(n+2), n=0, 1, 2, . . . has ensemble minimum distancegrowing linearly with block size.

Protographs for this AR4JA family are shown in FIG. 11. The thresholdsachieved by this family compared to the corresponding capacity limitsare also shown in FIG. 11. FIG. 12 shows bit and frame error ratesimulation results for LDPC codes with input block size k=4096 expandedfrom the rate-½ through rate-⅘ AR4JA protographs in FIG. 11, for whichprecoding was used to lower the decoding threshold.

The example in FIG. 2 illustrates that preceding is not essential forconstructing a code having both a low iterative decoding threshold andlinearly growing minimum distance. Instead, a high-degree variable nodecan be used to lower the iterative decoding threshold. In FIG. 2applicants start with a rate-⅘ code having variable node degrees 3except for one variable node of high degree 16. Then the rate-⅔ andrate-½ protographs in FIG. 2 are obtained via the check node splittingprocess described in [0023] to [0025].

It can be shown that it is in fact possible to design protograph-basedrate-½ LDPC codes with degrees at least 3 and maximum degree not morethan 20, with iterative decoding threshold less than 0.72 dB. One canstart with a rate-½, eight-node protograph with variable node degrees 3as shown in FIG. 13( a). The iterative decoding threshold for this codeis 1.102 dB. One of the nodes is then changed to degree 16 as shown inFIG. 13( b). The iterative decoding threshold for this code is 0.972 dB.Note that very little improvement is obtained by using one high-degreenode.

Now the connections from variable nodes to check nodes can also bechanged asymmetrically. After a few hand-selected searches theprotograph previously shown in FIG. 5 is obtained, which has threshold0.618 dB. Note that it is possible to obtain an even lower threshold(0.544 dB) if the number of nodes in the protograph is increased to 12and highest node degree is set to 18 as shown in FIG. 14.

A real-time field programmable gate array (FPGA) decoder has been usedto implement and test all of the protograph coding approaches describedin the present disclosure. The system, which is a universal decoder forsparse graph codes, operates using an in system programmable XilinxVirtexll-8000 series FPGA. Throughput on the order of 20 Mbps isachieved and performance plots of frame or bit error rate versus signalto noise can be computed using an integrated all white gaussian noisegenerator.

Accordingly, what has been shown among the four categories of codesconstructed in accordance with the methods described in the presentdisclosure are individual protograph-based LDPC codes with linearminimum distance and rate-compatible families of protograph-based LDPCcodes with linear minimum distance. While these individualprotograph-based LDPC codes and rate-compatible protograph-based LDPCcode families have been described by means of specific embodiments andapplications thereof, it is understood that numerous modifications andvariations could be made thereto by those skilled in the art withoutdeparting from the spirit and scope of the disclosure. It is thereforeto be understood that within the scope of the claims, the disclosure maybe practiced otherwise than as specifically described herein.

1. A digital communication coding method, comprising: providing a low-density parity-check (LDPC) code represented by a protograph, the protograph being provided with a linear minimum distance property, the protograph comprising at least one variable node with degree less than 3, and the protograph obtained by applying one or more operations in succession starting with a base protograph having all variable node degrees at least 3, each said one or more operations comprising a splitting of one check node of the base protograph or subsequent protographs into two check nodes and connecting said two check nodes with a transmitted degree-2 variable node.
 2. A digital communication coding method, comprising: providing a family of low-density parity-check (LDPC) codes of different rates but constant input block size, said family represented by a set of protographs of different rates, said protographs being provided with a linear minimum distance property, and said protographs obtained by applying one or more operations in succession starting with a base protograph having all variable node degrees at least 3, each said one or more operations comprising a splitting of one check node of the base protograph or subsequent protographs into two check nodes and connecting said two check nodes with either a transmitted degree-2 variable node or a non-transmitted degree-2 variable node.
 3. A digital communication coding method, comprising: providing a family of low-density parity-check (LDPC) codes of different rates but constant output block size, said family represented by a set of protographs of different rates, said protographs being provided with a linear minimum distance property, and said protographs obtained by applying one or more operations in succession starting with a base protograph having all variable node degrees at least 3, each said one or more operations comprising a connecting of two check nodes of the base protograph or subsequent protographs with either a non-transmitted degree-2 variable node or a non-transmitted degree-2 variable node set to zero.
 4. The digital communication coding method of claim 1, further comprising preceding one or more bits of the LDPC code.
 5. The digital communication coding method of claim 2, further comprising preceding one or more bits of each LDPC code of the family of LDPC codes.
 6. The digital communication coding method of claim 3, further comprising preceding one or more bits of each LDPC code of the family of LDPC codes. 